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When we calculate the standard deviation, we divide the root of the mean of squared deviations by root N, instead of N. The mean absolute deviation makes intuitive sense. When someone says that the mean absolute deviation (about the mean) of a series of numbers is "5 units", chances are that if you have take up a bunch of numbers from that series, they'll be more or less 5 units away from the mean (on average).

But with the SD, we can't say that. Given two standard deviations, we can tell which set of data is more consistent (less dispersed) by comparing the standard deviations, but on its own, saying that a data set has a standard deviation of "5" doesn't have the same intuitive sense as saying that the mean absolute deviation of a data set is 5.

So when someone says that the standard deviation of a data set is "5", what do they really mean?

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    $\begingroup$ Standard deviation is especially useful for normally distributed data, see this. For such data, SD allows you to give a precise meaning to "chances are that if you have take up a bunch of numbers from that series, they'll be more or less 5 SD units away from the mean". $\endgroup$ – Jean-Claude Arbaut Dec 6 '18 at 0:25

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